Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: From Approximation to Generalization: Integration, Exercise 29: Chapter review

Given that ${\int }_1^{4}f\left(x\right)dx=10 \text{and} {\int }_1^{3}f\left(x\right)dx=6$, find ${\int }_1^4\left(f\left(x\right)+4\right)dx$.

Let $f\text{'}\left(x\right)=4x^3+2$. Given that $f\left(2\right)=24$, find $f\left(x\right)$.

Find the area of the region enclosed by the graphs of the given functions $f\left(x\right)=x^4-5x^2 \text{and} g\left(x\right)=0.5e^x+1$.

Find the area of the region enclosed by the graphs of the given functions $f\left(x\right)=x^2-2x \text{and} g\left(x\right)=10x+x^2-3x^3$.

Find the equation of the tangent line to the graph of $f\left(x\right)=x^3 \text{at} x=1$.

The tangent line to the graph of $f\left(x\right)=x^3 \text{at} x=1$ intersects $f$ at one other point. Find the coordinates of this point.

Find the area of the region enclosed by the graph of $f\left(x\right)=x^3$ and the tangent to $f$  at $x=1$.

Consider the function defined by $f\left(x\right)=4x-x^3\text{ for }-2\le x\le 2$.

Find the exact value of  ${\int }_0^{2}f\left(x\right)\mathrm{d}x$